The New Exponentiated T-X Class of Distributions: Properties, Characterizations and Application

In this article, we introduce a new class of distributions called the New Exponentiated T-X family of distributions. A special sub-model of the proposed family, called a new exponentiated exponential-Weibull is considered in detail. Some structural properties associated with this new class of distributions are obtained. Certain characterizations of the proposed family are presented. Maximum likelihood estimators of the model parameters are obtained and Monte Carlo simulation study is conducted to evaluate the performances of these estimators. Finally, the importance of the new family is illustrated empirically via a real-life application.


1.
Introduction Speaking broadly, statistical distributions are widely used in modeling real phenomena of nature. Among these distributions, the Exponential, Rayleigh and Weibull are some of the important statistical models widely used in many applied areas. However, these distributions have a limited range of capability and thus cannot be applied in all situations. For example, although the exponential distribution is often described as flexible, its hazard function is constant, whereas, the Rayleigh has increasing hazard function only. The limitations of the classical distributions motivated the researchers to Using the T-X idea, several new classes of distributions have been introduced in the literature. Table 1 If T is the exponential random variable with parameter 0   , then its cdf is given by ( ) The density function corresponding to (2) is ( )

( )
; Fx  is the cdf of the baseline distribution which depends on the parameter vector . The pdf corresponding to (4) is given by Ahmad, Z., Ampadu, C.B., Hamedani, G.G., Jamal, F.
The main goal of this research is to introduce a new family of distributions, called the new exponentiated T-X("NE T-X for short) family of which the NE-Exponential X family discussed above is a special case. The generic form of the NE T-X family is introduced at the beginning of Section 3. We discuss a special sub-model of this family, capable of modeling with monotonic and non-monotonic hazard rates. For the special sub-model of the NE T-X family, a real life application is presented. The rest of this paper is structured as follows: In Section 2, a special sub-model of the proposed family is presented. Statistical properties of the proposed family are investigated in Section 3. Section 4 contains some useful characterizations of the proposed class. Section 5, provides estimation of the model parameters using maximum likelihood method. Section 6, provides analysis to a real data set. Simulation results are reported in Section 7. Finally, Section 8 concludes the article.

2.
Special Sub-Model Considering the cdf of the two-parameter Weibull model with shape parameter 0   and scale parameter 0,   given by ( )

3.
Mathematical Properties In this section, we provide some mathematical properties of the proposed class. The generic form of the cdf and pdf of the proposed class are given, respectively, by Skewness measures the degree of the long tail (towards left or right side). Kurtosis is a measure of the degree of tail heaviness. When the distribution is symmetric, S=0 and when the distribution is right (or left) skewed, S> 0 (or < 0). As K increases, the tail of the distribution becomes heavier.
Proof. Combine Corollary 3.1 with the definition of skewness given by Proof. Combine Corollary 3.1 with the definition of kurtosis given by   (8), if a random X follows the new exponentiated T-X class of distributions, then the Shannon entropy of X is given by where α>0, the random variable T has Shannon entropy, ηT, the random variable X has pdffX and quantile function QX, and given the random variable Q (say), µQ is the mean of Q, and ProductLog[·] is defined as before.
has density r(t), the result follows by noting that .
We get the desired result by using the expression immediately above in Given a random variable X with pdf f(x), the ordinary moments, for r∈ N, are given by However, if the random variable X in question has cdf F(x) and quantile function QX, then after the substitution u = F(x), the ordinary moments can be expressed as

4.
Characterizations This section deals with various characterizations of NE T-X distribution. These characterizations are based on a simple relationship between two truncated moments. It should be mentioned that for these characterizations the cdf may not have a closed form. Due to the nature of the proposed cdf, our characterizations may be the only possible ones. The first characterization result employs a theorem due to Glänzel (1987); see Theorem 4.1 in Appendix A. Note that the result holds also when the interval H is not closed. As shown in Glänzel (1990), this characterization is stable in the sense of weak convergence.
Conversely, if  is given as above, then Now, in view of Theorem 4.1, X has density (5). x ; where D is a constant. Note that a set of functions satisfying the above differential equation is given in Proposition 4.1 with D = 1/2. However, it should be also noted that there are other triplets ( ) 12 ,, qq satisfying the conditions of Theorem 4.4.
for some α > 0, where the random variable X has cdf ( )

Estimation
In this subsection, we determine the maximum likelihood estimates of the parameters of the NE T-X family. Let 12 , , · · · , k x x x be the observed values from the NE T-X distribution with parameters and . The total log-likelihood function corresponding to (5) is given by The partial derivates corresponding to (16), are given by  ; / 1;

Application
In this section, we illustrate the proposed method via analyzing a real data set taken from Saboor and Pogany (2016) representing the breaking strength of carbon fibers (in Gba). For the application section, we keep one parameter constant ( )  = and reduce the number of parameters to three. The comparison of the proposed distribution is being made with four other well-known extensions of the Weibull distribution. The cumulative functions of the competing models are: • Based on these measures, it is showed that the proposed model provides greater distributional flexibility. Corresponding to analyzed data set, the maximum likelihood estimates are provided in Table 2, whereas, the analytical measures are provided in Table  3.

7.
Simulation Study In this section, we assess the performance of the maximum likelihood estimators in terms of the sample size n. A numerical evaluation is carried out to examine the performance of maximum likelihood estimators for NEEW model (as particular case from the family). The evaluation of estimates is performed based on the following quantities for each sample size; the biases and the empirical mean square errors (MSEs) using the R software. The numerical steps are listed as follows: i. A random sampleX1, X2, . . . , Xn of sizes; n=30 and 50 are considered, these random samples are generated from the NEEW distribution by using inversion method. ii. Six sets of the parameters are considered. The MLEs of (Proposed) model are evaluated for each parameter value and for each sample size. iii. 1000 repetitions are made to calculate the biases and mean square error (MSE) of these estimators. iv.
Formulas used for calculating bias and MSE are given by v.
Step (iv) is also repeated for the other parameters ( ) ,, Empirical results are reported in Table (4). We can detect from these tables that the estimates are quite stable and are close to the true value of the parameters as the sample sizes increase.

Concluding Remarks
We have introduced a new function to extend the existing class of distributions. This effort leads to a new family of lifetime distributions, called the new exponentiated T-X family of distributions. General expressions for some of the mathematical properties of the new family are investigated. Maximum likelihood estimates are also obtained. There are certain advantages of using the proposed method like its cdf has a closed form and facilitating data modeling with monotonic and non-monotonic failure rates. A special sub-model of the new family, called the new exponentiated exponential Weibull distribution is considered and a real application is analyzed. In simulation study, the consistency and proficiency of the maximum likelihood estimators of the proposed model are also illustrated. The practical application of the proposed model reveal better fit to real-life data than the other well-known competitors. It is hoped, that the proposed method will attract wider applications in the area.